Problem: Determine how many solutions exist for the system of equations. ${-18x+3y = -30}$ ${y = 6x-3}$
Explanation: Convert both equations to slope-intercept form: ${-18x+3y = -30}$ $-18x{+18x} + 3y = -30{+18x}$ $3y = -30+18x$ $y = -10+6x$ ${y = 6x-10}$ ${y = 6x-3}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 6x-10}$ ${y = 6x-3}$ Both equations have the same slope with different y-intercepts. This means the equations are parallel. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Parallel lines never intersect, thus there are NO SOLUTIONS.